Low-temperature transport spectroscopy measurements on a suspended few-hole carbon nanotube quantum dot are presented, showing a gate-dependent harmonic excitation spectrum which, strikingly, occurs in the Coulomb-blockade regime. The quantized excitation energy corresponds to the scale expected for longitudinal vibrations of the nanotube. The electronic transport processes are identified as cotunnel-assisted sequential tunneling, resulting from nonequilibrium occupation of the mechanical mode. They appear only above a high-bias threshold at the scale of electronic nanotube excitations. We discuss models for the pumping process that explain the enhancement of the nonequilibrium occupation and show that it is connected to a subtle interplay between electronic and vibrational degrees of freedom.